On a problem of Rolewicz about Banach spaces that admit support sets
A nonempty closed convex set \(C\) in a Banach space \(X\) is called a \textit{support set} if for every \(c\in C\) there exists \(x^*\in X^*\) such that \(x^*(c)=\inf\{x^*(x):\;x\in C\}<\sup\{x^*(x):\;x\in C\}\). In [\textit{S.\,Rolewicz}, ``On convex sets containing only points of support'', Commentat.\ Math., Spec.\ Vol.\,I, dedic.\ L.\,Orlicz, 279--281 (1978; Zbl 0385.46009)], it was proved that there are no separable support sets, and it was asked whether every nonseparable Banach space contains a support set. In the present paper, it is shown that this is not the case, by providing an example of a nonseparable Banach space (consistent with the axioms of ZFC) of type \(C(K)\) that contains no support set. It was proved in [\textit{S.\,Todorčević}, ``Biorthogonal systems and quotient spaces via Baire category methods'', Math.\ Ann.\ 335, No.\,3, 687--715 (2006; Zbl 1112.46015)] that it is consistent that any nonseparable Banach space contains an uncountable biorthogonal system. It follows that the nonexistence of the example in the present paper is also consistent with the axioms of ZFC, so Rolewicz's problem turns out to be undecidable. The example provided here is, as mentioned, a space of type \(C(K)\), and it is hereditarily Lindelöf in the weak topology. It does not have any nonseparable subspace nor any nonseparable quotient of type \(C(M)\) for \(M\) dispersed compact, so, unlike Kunen's space, it is not Asplund. Indeed, every nonseparable \(C(K)\) space for \(K\) dispersed admits a support set, as follows from results in [\textit{A.\,J.\thinspace Lazar}, ``Points of support for closed convex sets'', Ill.\ J.\ Math.\ 25, 302--305 (1981; Zbl 0437.46008)]. The construction of the example requires the study of the properties of the Boolean algebra of clopen sets of the compact set \(K\) (\(K\) is totally disconnected, so it is the Stone space of a Boolean algebra; the space \(K\) is obtained somehow from the Cantor set by a splitting procedure). This is done in Section~2. The consistency of the existence of the algebra is done by using the method of forcing. This is the purpose of Section~3. In Section~4, right-separated sequences in the dual ball of the space \(C(K)\) with the weak\(^*\) topology are analysed. To show that the example provided has no support sets, the author proves that it has no uncountable semibiorthogonal sequences of length \(\omega_1\), a concept introduced in [\textit{J.\,Borwein} and \textit{J.\,Vanderwerff}, ``Banach spaces that admit support sets'', Proc.\ Am.\ Math.\ Soc.\ 124, No.\,3, 751--755 (1996; Zbl 0840.46005)], for it is proved there that the existence of a semibiorthogonal sequence of length \(\omega_1\) characterizes Banach spaces with support sets. The author mentions that another example of a nonseparable Banach space having no support set has been provided, roughly at the same time, by Todorčević. This is a remark in \textit{S.\,Todorčević} [op.\,cit.]\ and the example comes from [\textit{M.\,Bell, J.\,Ginsburg} and \textit{S.\,Todorčević}, ``Countable spread of \(\exp Y\) and \(\lambda Y\)'', Topology Appl.\ 14, 1--12 (1982; Zbl 0502.54004)].
scientific article; zbMATH DE number 5618753
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